Continuation of a BPC curve in an equivariant system and understanding BPC options

[Evgenij Gr.]

Hello!

In the system that I am studying, a Branch Point of Cycles bifurcation occurs during the single-parameter continuation of stable limit cycle. Since the system possesses discrete symmetries (permutations of some phase variables, also reflections), it is quite likely that in this case BPC might have lower codimension. Is it possible to compute this curve in Matcont, either directly or by using some workaround?

The second question is related to the initializers that are available, when selecting BPC point. There are four options: init_BPC_LC, init_LC_LC, init_BPC_LPC and init_BPC_BPC. I struggle to find the explanation for these options and what exactly do they continue. My primary goal right now is to compute the BPC curve on two-parameter plane since that curve is a boundary of stability for a periodic solution that is interesting for me.

Thanks in advance for your help!

[Alois Steindl]

Hello,
as far as I know, MatCont doesn't deal with symmetries at all.
One possible method would be to introduce symmetry breaking terms, the coefficients of these terms would be considered as imperfections.
Look at the pizchfork bifurcation
x'=x^3-lambdax
with mirror reflection symmetry x-> -x. The branch point at (0,0) can be seen as regular branchpoint in the perturbed system
x'=x^3-lambdax+alpha.

While this method is cumbersome, when you are only interested in the symmetric case, it pays off when you investigate the real system, which always will have broken symmetries.
Good luck
Alois